Discrete Double-Bracket Flows for Isotropic-Noise Invariant Eigendecomposition

We study eigendecomposition on $SO(n)$ under streaming observations $C_k=C_{\mathrm{s}\mathrm{i}\mathrm{g}}+{\sigma }_k^{2}I+E_k$, where the isotropic background ${\sigma }_k^{2}I$ may be time-varying and arbitrarily large. Standard algorithms couple their stability to $\parallel C_k{\parallel }_2\approx {\sigma }^2$, forcing step sizes, contraction rates, and iteration counts to degrade with the noise floor. We observe that ${\sigma }^{2}I$ lies in the center of the matrix algebra and therefore should never enter the eigenspace dynamics. We construct a discrete double-bracket flow whose skew-symmetric generator $\mathrm{\Omega }=[A,\mathrm{diag}(A)]$ operates in the tangent Lie algebra $\mathfrak{s}\mathfrak{o}(n)$, where scalar multiples of the identity vanish by antisymmetry. The resulting trajectory, Lyapunov function, and maximal stable step size ${\eta }_{\max }=1\mathrm{/}{L}_C$ depend exclusively on the trace-free signal $C_e$ – achieving pointwise, pathwise ${\sigma }^2$-invariance. We establish input-to-state stability with a noise ball governed solely by trace-free perturbations, prove global convergence via strict-saddle geometry and a discrete Łojasiewicz argument, and extend the framework to top-$k$ eigentracking on the Stiefel manifold $\mathrm{St}(k,n)$ at cost $k$ matrix-vector products per step.